## Loving math since childhood

Oshima has loved math since he was in elementary school. He would solve math puzzles and read math books for children even after he got home from school.

“I think my parents had a big influence on me. My father was a math researcher, and my mother taught high school math. Still, they never told me to do math. I think I naturally became interested in it because math books and calculation notes were all around me. When my parents noticed that I was interested, they bought me books and created an environment where I could become familiar with math. I remember my father used to make puzzle-like problems for me. For example, he would ask me to half the area of a triangle made up of matchsticks using a few additional matchsticks. These were problems that even a child could understand. However, rather than having only one answer, the problems allowed for thinking and trial and error. It was fun. Even if I did not get an answer right away, I could still make small discoveries if I thought hard about the problem.”

The young Oshima was fascinated by arithmetic and mathematics in elementary, middle, and high school.

“At the time, I did not think about doing mathematics research in the future at all. I just enjoyed thinking about mathematical problems and being exposed to mathematics. When I read books on arithmetic and mathematics, even if at first I did not understand what I was reading, I gradually started to gain an understanding after taking the time to think. The process was fun, and that feeling has not faded.”

When Oshima was a university student and decided to become a mathematician, he chose the field of the “representation of Lie groups.”

“The field of representation theory bridges different areas of mathematics and is connected to all areas of mathematics. Research in this field can lead to solving problems in unexpected ways.”

Oshima describes his research, but some explanations are needed. What is a “group”? And what is “representation,” a concept that sounds like it belongs to the humanities?

## What are sets? What are groups?

“In the abstract, a set and the operations that can be performed on its elements define a group,” Oshima explains.

“Group” and “set” are everyday English words. So, they are not that scary. A set could be, for example, the set of types of chocolates, which could include the set (subset) of types of dark chocolates. But how is a group different from a set?

“The rules of operations, how we are allowed to manipulate the members of a set, are fixed. If the set and the operations performed on its elements satisfy certain conditions, we call it a group.”

In other words, the set of types of chocolate becomes a group when we do “operations” on its members... Admittedly, using the chocolate metaphor for further explanations is difficult.

Operations are manipulations such as addition and multiplication. The results of the operations must be elements of the set to satisfy the conditions. For example, the “group of a set of integers with addition” is defined by the set of all integers and the operation of addition. It seems like a matter of course that adding integers to integers results in integers. So why is it necessary to specify this as a condition? Would addition within the set of natural numbers also qualify as a group? The answer is no because “addition within the set of natural numbers” violates one of the conditions.

A group needs to satisfy three conditions. The first condition is that the associative property “(a+b)+c=a+(b+c)” must hold. The second condition is that there is an identity element, a number that leaves the other elements of the set unchanged after the operation. For example, in the case of addition, this is the number “0” (0 + 3 = 3). The third condition is that there is a number for each element (called the inverse element) for which the result of the operation is the identity element. If 0 is the identity element, then for the number 2, “-2” will be the inverse element because adding it results in 0, the identity element. More generally, in the case of addition, for any integer, there is a negative integer that, when added, results in 0. This means that there is an inverse element for all elements, and the third condition is satisfied. From this, it follows that the “set of integers with addition” is a group. The set of natural numbers does not include negative numbers. Therefore, inverse elements do not exist in this set, and the “set of natural numbers with addition” is not a group. Not any set can be a group, after all.

This is merely the definition of groups. Although it is simple, a mathematically rich world lies behind these peculiar sets called groups. Oshima's research subject, “group representation,” is about “representing” groups, sets that satisfy three simple conditions, in the language of mathematics, such as linear algebra, a branch of mathematics even university students study, and revealing their true potential.

## Group representation is the “language” of symmetry

“The definition of group representation is a representation in which each element in the group corresponds to a cell in a matrix so that the group's product corresponds to the matrix's product. One goal of group representation is to try to understand abstract groups by putting them in more concrete terms, matrices.”

Matrices, usually taught as a part of undergraduate mathematics, are rows and columns of numbers and symbols. For example, if the first row is [1 2] and the second row is [3 4] that makes up a single matrix of two rows and two columns. Then, matrices can be added to and multiplied by each other (although the rules are slightly complicated). Group representation represents all elements of a particular group in a matrix, which allows mathematicians to reveal the group's properties. And, Oshima says that group representation naturally emerges where there is “symmetry.”

“It is hard to tell from the definition alone, but groups and representations are mathematical concepts that describe symmetries. In other words, when you want to capture the symmetry that appears in various mathematical figures, functions, and expressions, or when you want to use symmetry to investigate something, you need groups and group representations.”

An everyday example should provide a clearer picture. Let us consider the symmetry of a shape. An isosceles triangle is symmetrical about the line, called the “line of symmetry,” running through its apex and the center of its base. A circle is symmetrical about its diameter. Moreover, a circle has rotational symmetry: it does not change shape even when rotated. Symmetry is relevant not only in the world of geometry but also in the world of elementary particles, where there is a symmetry of time; if time was reversed, that would not affect the physics at that level. There is indeed a variety of symmetries.

“For example, a circle is a group. No matter how you rotate a circle, its shape remains the same. So, each rotation makes up an element of the group. For example, rotating a circle by 30 degrees is an element of the group, and rotating it by 60 degrees is also an element. Rotating by 360 degrees returns the shape to its starting point, which is the same as rotating it by 0 degrees. Then, the circle is complete. Each transformation is an element, and together they satisfy the conditions of groups. The transformation of a circle can be represented by a matrix with two rows and two columns. This is the representation of the group called a circle.”

Oshima's research subject is the Lie group, one of the many existing groups. The Lie group was developed in the 19th century by the Norwegian mathematician Sophus Lie. Oshima says that the distinctive feature of the Lie group is that it deals with continuous symmetry

“Rotation is a type of continuous symmetry. Rotation was used to create the circle in the previous example, so a circle is a Lie group. Three-dimensional rotations on a sphere form the “rotation group.” The totality of three-dimensional rotations is also a Lie group. In such a way, a Lie group appears when a transformation can be done continuously, i.e., when the resulting shape is always the same after continuous transformation. Therefore, studying Lie groups helps to analyze such continuous symmetries.”

Oshima received the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology Young Scientists' Award for his work titled “The unitary representations of real reductive Lie Groups” and is internationally recognized as a young researcher in Lie group representation.

## Mathematicians might look like they are taking a break when working

The representation theory of groups plays a significant role in physics... so much so that some say the development of the study of Lie groups around the 1950s was a reaction to the questions posed by quantum mechanics.

“Symmetry appears in various natural sciences. So, to mathematically quantify it, you inevitably need group representation. Still, it is fascinating to see how something mathematicians deem beautiful or important from a mathematical perspective and formulate theorems about is used in a different context elsewhere, such as in physics. Mathematics may be, in a sense, a human-made world of ideas, but it is connected to the real world in deep and unexpected ways. I find that incredible.”

Oshima gave the Fourier transform as an example of the interesting connection between the real world and group theory. The Fourier transform is a function that plays an important role in headphone noise-canceling technology.

“For example, the Fourier transform (Fourier analysis) is used to decompose a sound into simpler sound waves or to decompose light, similarly to a prism, into simpler wavelengths of light. In other words, the Fourier transform separates overlapping waves of various wavelengths into multiple simple waves. The representation of Lie groups can capture such simple waves because they can be viewed as highly symmetric.”

The significance of mathematics is not restricted to group theory; it is impactful in the real world. Mathematicians do not look through astronomical telescopes, collide subatomic particles, or analyze DNA with sequencers; all they need is a pen and paper. Oshima elaborates.

“Sometimes, I am at my desk, writing complex calculations on paper. However, when I think about mathematical problems more loosely, it may appear to an outside observer that I am taking a break. But taking time to think freely is important for research.”

What kind of discipline is mathematics, a world that exists only in people's minds?

“There is mathematics as the language of natural science. There is also “artistic” mathematics, the mathematics that mathematicians find beautiful when they look at theories and theorems. Also, in mathematics, like in the natural sciences, there are what I would like to call “phenomena.” So, there is mathematics born out of the desire to know and working on why such phenomena occur. Discovering, observing, and understanding these mysterious “phenomena” is mathematical research. I want to experiment and find interesting “phenomena,” similarly to a scientist.”

However, the word “phenomenon” sounds strange in the context of mathematics.

“Yes, it does sound a little strange. I think that deep inside, people think of mathematics as a natural phenomenon that exists outside and not just in their minds. It is difficult to say whether the world of mathematics is “real” outside the human mind because it is a philosophical question, but it is tempting to use the word “phenomenon.”

## Math is full of things we have yet to understand

Oshima says he enjoys his daily research very much.

“Someone once told me that mathematics is similar to when you walk into a dark room with no lights on, you have to fish around to orient yourself, but once you finally find the switch and turn on the light, everything suddenly becomes clear. In mathematics, when you do not understand something, you do not understand it at all. Then, when you finally take a step forward, you suddenly gain such a clear understanding that you fail to see what the difficulty was in the first place. But then a new problem you do not understand appears instantly (laughs). Nonetheless, I have fun doing mathematics even when I struggle to figure things out.”

From the outside, it might appear that the study of mathematics has mostly been completed and organized and that there is no longer any unexplored territory.

“I think that is not the case. There are many things that I do not understand. It is like a never-ending landscape of mountains waiting to be climbed. And I think many mathematicians feel similarly to the way I do. Little by little, through the efforts of many different people, we are increasing the number of things we know. And the more we move forward, the more we discover many things we do not yet understand. Personally speaking, there are many things I do not understand even within my current field of research. But I want to understand. That is my immediate goal.”

Mathematics is still full of things we have yet to understand, so Oshima says there are many worthy challenges young researchers can take on.

“In mathematics, finding something new and interesting is not easy. It is not only about solving problems but also about finding problems and new mathematical phenomena. But that is what makes it so rewarding.”

In his private life, he is a father of two children.

“My children are still young, so my days are very lively. Is math in the picture already? No, they do not seem interested in it now (laughs).”

※Year of interview:2024

Interview/Text: OTA Minoru

Photography: KAIZUKA Junichi